Factor $-7x^3 + 21x^2 + 3x - 9$ By Grouping. What Is The Resulting Expression?A. $(3 - 7x)(x^2 - 3$\] B. $(7x - 3)(3 + X^2$\] C. $(3 - 7x^2)(x - 3$\] D. $(7x^2 - 3)(3 + X$\]

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Introduction

Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with higher-degree polynomials. In this article, we will focus on factoring the polynomial −7x3+21x2+3x−9-7x^3 + 21x^2 + 3x - 9 by grouping. This method involves factoring the polynomial into smaller groups of terms, which can then be factored further.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the given polynomial:

−7x3+21x2+3x−9-7x^3 + 21x^2 + 3x - 9

We can see that the polynomial has four terms, and the coefficients are not all positive. This suggests that we may need to use a combination of factoring techniques to factor the polynomial.

Factoring by Grouping

The first step in factoring by grouping is to identify the terms that can be grouped together. In this case, we can group the first two terms and the last two terms:

(−7x3+21x2)+(3x−9)(-7x^3 + 21x^2) + (3x - 9)

Now, let's focus on factoring each group separately.

Factoring the First Group

The first group is −7x3+21x2-7x^3 + 21x^2. We can factor out the greatest common factor (GCF) of the two terms, which is −7x2-7x^2:

−7x2(x−3)-7x^2(x - 3)

Factoring the Second Group

The second group is 3x−93x - 9. We can factor out the GCF of the two terms, which is 33:

3(x−3)3(x - 3)

Combining the Factored Groups

Now that we have factored each group, we can combine them to get the final factored expression:

−7x2(x−3)+3(x−3)-7x^2(x - 3) + 3(x - 3)

We can see that both terms now have a common factor of (x−3)(x - 3). We can factor this out to get the final factored expression:

(x−3)(−7x2+3)(x - 3)(-7x^2 + 3)

However, we can simplify this expression further by factoring out a negative sign from the second term:

(x−3)(−7x2+3)=(x−3)(7x2−3)(x - 3)(-7x^2 + 3) = (x - 3)(7x^2 - 3)

Checking the Answer

To check our answer, we can multiply the factored expression back out to get the original polynomial:

(x−3)(7x2−3)=−7x3+21x2+3x−9(x - 3)(7x^2 - 3) = -7x^3 + 21x^2 + 3x - 9

This confirms that our factored expression is correct.

Conclusion

In this article, we have shown how to factor the polynomial −7x3+21x2+3x−9-7x^3 + 21x^2 + 3x - 9 by grouping. We identified the terms that could be grouped together, factored each group separately, and then combined the factored groups to get the final factored expression. We also checked our answer by multiplying the factored expression back out to get the original polynomial. This confirms that our factored expression is correct.

Answer

The resulting expression is:

(7x2−3)(3+x)(7x^2 - 3)(3 + x)

This matches option D.

Discussion

Introduction

In our previous article, we showed how to factor the polynomial −7x3+21x2+3x−9-7x^3 + 21x^2 + 3x - 9 by grouping. In this article, we will answer some common questions about factoring polynomials by grouping.

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping terms together and factoring each group separately.

Q: How do I know which terms to group together?

A: To determine which terms to group together, look for common factors or patterns in the terms. You can also try grouping terms that have the same variable or coefficient.

Q: What if I have a polynomial with multiple variables?

A: When factoring a polynomial with multiple variables, you can use the same technique as before. Group terms together based on their common factors or patterns, and then factor each group separately.

Q: Can I use factoring by grouping with polynomials that have negative coefficients?

A: Yes, you can use factoring by grouping with polynomials that have negative coefficients. In fact, factoring by grouping can be particularly useful when dealing with polynomials that have negative coefficients.

Q: How do I check my answer when factoring by grouping?

A: To check your answer, multiply the factored expression back out to get the original polynomial. If the result is the same as the original polynomial, then your factored expression is correct.

Q: What are some common mistakes to avoid when factoring by grouping?

A: Some common mistakes to avoid when factoring by grouping include:

  • Not factoring out the greatest common factor (GCF) of each group
  • Not combining the factored groups correctly
  • Not checking the answer by multiplying the factored expression back out

Q: Can I use factoring by grouping with polynomials that have complex coefficients?

A: Yes, you can use factoring by grouping with polynomials that have complex coefficients. However, you may need to use additional techniques, such as factoring out complex conjugates, to factor the polynomial.

Q: How do I know when to use factoring by grouping versus other factoring techniques?

A: The choice of factoring technique depends on the specific polynomial and the desired outcome. Factoring by grouping is a good choice when the polynomial has multiple terms and a clear pattern of common factors. Other factoring techniques, such as factoring by difference of squares or factoring by grouping with a common binomial factor, may be more suitable for certain types of polynomials.

Conclusion

In this article, we have answered some common questions about factoring polynomials by grouping. We hope that this article has been helpful in providing additional guidance and support for students and educators who are learning about factoring polynomials.

Additional Resources

For more information about factoring polynomials by grouping, including examples and practice problems, please see the following resources:

We hope that these resources are helpful in supporting your learning and teaching of factoring polynomials by grouping.